I'm lead to believe that these types of descriptors are subjective, but given the fact that many people use the same words for similar sounds, there must be something more objective to grasp on to.

What are the quantitative/qualitative differences between instruments that allow their tones to be described as "bright," "dark," "round," "edgy," etc.? Is there a shared qualitative basis between guitars, amps, pedals, or even different types of instruments that lead to the same tone descriptor?

On the quantitative side, are there particular waveform shapes/sizes that are assigned particular descriptors? For example, can I take a "fat" sound, adjust it's waveform, and end up with a "crunchy" sound (replace with any 2 descriptors)?

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Those terms mostly describe the frequency characteristics of a given sound and how the person feels about those characteristics. For instance, an emphasis on lower frequencies can be characterised positively as "warm" or "mellow", or negatively as "muddy". Likewise, an emphasis on higher frequencies can be characterised positively as "bright" or "crisp", or negatively as "harsh". Other terms might describe the presence or lack of compression ("punchy", "tight", "loose"), distortion ("edgy", "crunchy", "smooth") or other type of manipulation of the source audio ("coloured", "transparent"), while yet others are completely ambiguous ("round").

Which term corresponds to which type of sound is, as you've been correctly led to believe, largely subjective and depends a lot on context (for instance, a subwoofer could be described as muddy compared to full-range speakers, but bright and punchy compared to other subwoofers), the listener's personal taste and listening ability, and so on. Granted, some words are more likely to be used to describe certain types of sounds (as shown in the previous paragraph), most likely due to existing meanings those words already possess that have simply been adapted to describe sounds, but there are no objective standards.

As for waveforms, it's not as simple as you describe. Not all of the terms you listed are mutually exclusive, and a waveform can (and will) contain information about different aspects of a sound - its frequency response, amplitude, dynamic range, total harmonic distortion, phase, etc. While the things you describe could be done with a simple sine wave (for instance, increase frequency to go from "fat" to "bright" or clip the tips to go from "smooth" to "crunchy" or "edgy"), with any real audio the waveform is going to be too complex to use as a basis for determining or changing the tonal qualities of a sound. Instead, look into tools actually used for this, like equalisers, compressors, limiters, amplifiers, filters, etc.

The fundamental problem you are struggling with is that human language is inadequate to describe sounds and the timbres of instruments. This goes back to the maxim "Writing about music is like dancing about architecture."

Any time anybody tries to create written descriptions of sounds, that person has to apply their own subjective vocabulary.

So no, there is nothing especially quantitative or qualitative about a written description of a sound. You simply have to hear the sounds yourself to know exactly what is being described verbally. In the absence of the ability to do that, you have to try to glean something from a writer's attempt to describe sounds using human language.

I believe this is a form of synasthesia, relating features of the sound to the poetry of words. For a very compelling example, see this TED talk (also on netflix in the Brave Neuro World collection). Ramachandran presents two letters from a "Martian" language, and most of the audience has the same opinion about which letter is which. For some people, this descriptive ability is innate, for others it is learned.

The basic characteristics like bright or mellow are attributable to the envelope of the frequency response. You can easily verify this by experimenting with an equalizer.

Equalization can fatten sounds or make them more brittle.

Note that equalization occurs between distortion stages as well as before and after.

One thing that you cannot do with an equalizer is to change the sound of a pickup that is close to the neck to a bridge sound, or vice versa.

What's going on there is a frequency-dependent filtering, so it is not a simple transfer function in the complex frequency domain. (It can be achieved with digital signal processing. Atsushi Hoshiai of Roland Corp. has a patent on this.)

As you know, the characteristics of the pickup, and its placement, still manifest through through distortion. So using equalization anywhere in the signal chain, you will not turn a distorted bridge humbucker tone into a distorted neck single coil tone.

Nevertheless, you can create an amazing variety of tones with equalization.